Yesterday we worked on finding ways to convert from polar ordered pairs to rectangular (or vice versa). Today's lesson will focus on how to find the distance between two points that are given in polar form. I really like this lesson because it is an extension of yesterday and gives students more practice in visualizing polar points in the plane. Students also use past geometric concepts (distance formula and Law of Cosines) in an entirely new coordinate system.
I begin the lesson by showing students slide #2 of the PowerPoint and asking them to find the distance between each of the three pairs of points. We will remind ourselves that these ordered pairs are all written in polar form and recap what each coordinate represents in polar form. I will have students work in their table groups and I will give them about 10-15 minutes to see how far they can get. I stress that there are multiple ways to find the distance between points; if they find an answer for the first pair of points, they should try to use a different method to find the distance for the second pair.
As I walk around the room, here are some hints I will give if students are stuck.
After students have had ample time to work, I will select students to share their work with the class. Below is the order I will hand pick students to share their work with the entire class. Obviously it will depend on the solution strategies I see, but these are usually bound to be present. If one of the solution strategies is not present, you can always get your students started and see if they can finish it.
1. Using the distance formula incorrectly - This overgeneralization of the distance formula is usually present. A student will simply plug the r and θ in for x and y in the rectangular distance formula. Usually a student will catch their mistake while talking it over with their table, but I still think it is really important to talk about why this does not work. Students will need continuous reinforcement that the first coordinate in rectangular form means something completely different than the first coordinate in polar form.
2. Using the distance formula correctly - This is usually the most common strategy for my students. They will convert each of the two points to rectangular form (like we did yesterday) and then plug the new values into the distance formula. I like how the student whose work is shown here left the ordered pairs in (rcosθ, rsinθ) form. This may help students find a general formula for the distance between any two polar points.
3. Using the Law of Cosines - This strategy requires a diagram and the corresponding algebraic work. Again, if no students used this strategy, simply connecting each of the two points to the origin may get them to see a triangle and think about tools they can use.
Finally, the last pair of polar points from the PowerPoint contains all variables, so we will discuss how we can generalize the techniques we came up with to work for any two points in polar form.
Using the distance formula is pretty straightforward, but the Law of Cosines may have some interesting conversations. I discuss in the video below.
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After this discussion, I will give students this worksheet and have them work on it for homework.